In Exercises 11–26, determine whether each equation defines y as a function of x. 4x = y²

Verified step by step guidance

1. Step 1: Recall the definition of a function. A function is a relation where each input (x) corresponds to exactly one output (y). To determine if the given equation defines y as a function of x, we need to check if each x-value produces a unique y-value.
2. Step 2: Start with the given equation: $4x\; =\; y^2$. Rearrange it to isolate y. Divide both sides of the equation by 4 to get .
3. Step 3: Solve for y by taking the square root of both sides. Remember that taking the square root introduces both a positive and a negative solution. This gives $y\; =\; ext\{\pm \}\; ext\{\surd \}(4x)$.
4. Step 4: Analyze the result. Since y can take two values (positive and negative) for a single x-value, the equation does not define y as a function of x. A function must have only one output for each input.
5. Step 5: Conclude that the equation $4x\; =\; y^2$ does not define y as a function of x because it fails the vertical line test, which states that a vertical line drawn through any x-value should intersect the graph at most once.

Key Concepts

Function Definition

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, no two ordered pairs can have the same first element with different second elements. This concept is crucial for determining if an equation defines y as a function of x.

Vertical Line Test

The vertical line test is a visual way to determine if a curve is a function. If any vertical line intersects the graph of the relation more than once, then the relation is not a function. This test helps to quickly assess whether an equation can be expressed as y in terms of x without ambiguity.

Solving for y

To determine if an equation defines y as a function of x, it is often necessary to solve the equation for y. In the case of the equation 4x = y², rearranging it to express y in terms of x reveals whether y can take on multiple values for a single x. This step is essential for understanding the relationship between the variables.